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Math 10 Ms. Albarico 1. 2. 3. 4. 5. Introduction to Trigonometry Review on Types of Triangles Labelling Triangles Trigonometric Ratios Solving Word Problems Related to Trigonometry Introduction Triangles Around Us For this activity, you'll need: • a piece of paper, pencil • a ruler, protractor • some coloured pencils or pens What To Do: 1) Using your pencil and ruler, draw some straight lines on your piece of paper to make an interesting pattern. You can draw as many or as few as you like. • 2) Using your coloured pencils or pens, decorate all the threesided shapes in some way. You could colour them all in using a particular colour or you could cover them with a special design or pattern When you finish your work, you have to explain your design by answering this questions: • • • • • What is your pattern all about? Can you describe what you see in your own pattern? Can you find any shapes which have three sides? How about any with four sides? Which shape or shapes have the least sides? • What are the measures of their angles? • What is the total sum of their angles? Students are expected to: • M04.01 Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios. • M04.02 Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given a cute angle in the triangle. • M04.03 Solve right triangles, with or without technology. • M04.04 Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem. • M04.05 Solve a problem that involves indirect and direct measurement, using the trigonometric ratios, the Pythagorean theorem, and measurement instruments such as a clinometer or metre stick. Vocabulary perpendicular parallel sides angle triangle congruent similar dilate sail navigate approach Key Terms Primary Trigometric Ratios Sine Ratio Cosine Ratio Tangent Ratio Pythagorean Theorem Angle of Inclination Angle of Elevation Angle of Depression What is a TRIANGLE Triangles classified according to their Sides: • Scalene Triangle. A scalene triangle that has no equal sides. • Isosceles Triangle. An isosceles triangle is a triangle that has two equal sides. • Equilateral Triangle. An equilateral triangle is a triangle that has three equal sides. Identify the following triangles: Triangle B Triangle A Triangle C Triangles classified according to their Angles: • Right Triangle. A right triangle has a 900 angle. • Obtuse Triangle. An obtuse triangle has one angle that has bigger than 900 angle. • Acute Triangle. In an acute triangle, all angles has less than 900 angle. Identify the following triangles: Triangle B Triangle A Triangle C Names of the sides: – the HYPOTENUSE – the OPPOSITE side – The ADJACENT side • A REFERENCE ANGLE must be selected before you start labeling the sides of the right angle • It can be either of the two acute angles. • The HYPOTENUSE is the longest side of the right triangle. • The OPPOSITE SIDE is the side across the 900 angle. • Based from our drawing, which is our opposite side? • The ADJACENT SIDE is the side next to our reference angle. • The ADJACENT SIDE is the side next to our reference angle. • Given the same triangle, how would the sides be labeled if we choose the other acute angle as our reference angle? • Will there be any difference? • Which side does not change? • Primary Trigonometric Ratios – constant values based on the ratios of sides for particular angles in right-angled triangles. Sine, Cosine, and Tangent are called primary trigonometric ratios. • tan X – a constant value based on the ratio of the length of the side to a chosen angle X in a right triangle. • sin X – a constant value based on the ratio of the length of the side opposite to a chosen angle X to the length of the hypotenuse in a right triangle. • cos X – a constant value based on the ratio of the length of the side adjacent to a chosen angle X to the length of the hypotenuse in a right triangle. 5 3 x 4 3 5 x 4 5 3 x 4 opp sin x hyp o adj cos x hyp Take the first letter of each word. o opp tan x adj o S O H C A H T O A Note: Given Ratio of sides Angle, side Looking for Use Angle measure SIN-1 COS-1 TAN-1 Missing side SIN, COS, TAN Calculator Commands Reminder Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit Calculator Commands Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit Write the ratio for sin A B Sin A = a c c Write the ratio for cos A a C b A Cos A = b c Write the ratio for tan A Let’s switch angles! Find the sin, cos and tan for Angle B. Tan A = a b 3 5 x 4 To solve for Angles: tan x o opp adj C hyp’ Now we need to look at the two ratios involving the hypotenuse: sin xo = Opposite Hypotenuse cos xo = Adjacent Hypotenuse Opp’ xo A Adj’ B • For Trigonometric Inverse Functions: Press 2nd, use SIN for SIN-1 COS for COS-1 TAN for TAN-1 Name “say” Abbreviation Abbrev. Ratio of an angle measure Sine Cosine Tangent Sin Cos Tan sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side • What are the three sides of a right angle triangle? • In your calculator, what function will we use to find angle measures? • How can you remember easily the trigonometric ratios? Open your textbook to page 70. Find an angle that has a tangent (ratio) of 3 C 3cm B 4 Round your answer to the nearest hundredth degree. 4cm A Process: I want to find an ANGLE. I was given the sides (ratio). Tangent = opposite adjacent Solution: TAN-1(3/4) = 36.87° CLASS BOARD WORK INDIVIDUAL CLASS WORK • #10-14 ON page 76. HOMEWORK • #15-19 ON page 75-76. ASSIGNMENT 2 ASSIGNMENT 2 ASSIGNMENT 2 CLASS BOARD WORK INDIVIDUAL CLASS WORK • #3-8 ON page 82. HOMEWORK • #10-16 ON page 83. ASSIGNMENT 3 • - By partner, bring the following: Piece of string Heavy object (stopper) Needle Drinking straw (must be at least 15cm long) Read pages 85-86. How are you going to measure an inaccessible height? ASSIGNMENT 3 1) Students will create their own clinometer by pair. 2) Teacher will assign which inaccessible height to measure. 3) Students answer E-G on page 86 and Assess Your Understanding #1-3 Resources: • (PEARSON CANADA) Foundations and Pre-Calculus Math 10 • (NELSON CANADA) Mathematical Modelling Book 1 • Polyhedron figures For Tutorial • http://www.sd43.bc.ca/Resources/ParentRes ources/math/10/Pages/Trigonometry.aspx • https://www.mathsisfun.com/rightangle.htm l References: • https://www.mathsisfun.com/geometry/trian gles-interactive.html • http://www.math10.ca/lessons/measurement /trigonometryOne/trigonometryOne.php • mathworld.wolfram.com